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GEIR LOLANI"

TECHNISCHE UN1VERSIMT Laboratorium voor Scheepshydrornechanic8 Archlef Mekelweg 2, 2628 CD De Tel.. 1,15 - Man For 016 713 .735

CURRENT FORCES ON AND

FLOW THROUGH FISH FARMS

DOKTOR INOENIORAVHANDLING 1991:16 INSTITUTE' FOR MARIN HYDRODYNAMIKK

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Current forces on and flow through fish

farms.

by

Geir Loland

Division of Marine Hydrodynamics

The Norwegian Institute of Technology

1991

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ABSTRACT

A method for calculation of current forces on fish farming structures is developed.

Comparison with model test shows that the model reproduces the main features of the flow problem. In general, the derived model gives drag forces within the range of 90 to 120% of the measured drag forces. The only calibrated factor in the model is the drag coefficient for a single net panel. The model is based upon the assumption that the net cage can be divided into several net panels, and that the drag force on a net cage is given as the sum of the drag forces on the different net panels. The drag force on each net panel isgiven by equilibrium

between drag force, weight of sinkers and deformation of the net panel.

The wake behind a screen is investigated with the aid of thelinearized wake equations. The solution is matched with a far field solution. The wake is also analyzed by using a source

model. Both these approaches show good agreement with measurements of the velocity in the wake.

Based on the analysis of the wake flow, we have derived a relation between the velocity

reduction factor and the drag coefficient. The velocity reduction factor is defined as the ratio between the velocity in the wake behind a screen and the free flow. The flow velocity within

the cage system will be a function of the net panel upstream. The velocity is reduced in accordance with the velocity reduction factor when passing a netpanel.

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ACKNOWLEDGEMENT.

This dr.ing work is carried out as one part of the research program "Current forces on and

flow through fish farming structures'' funded by The Royal Norwegian Council for Scientific and Industrial Research (NTNF), Norwegian Marine Technology Research Institute (MARINTEK A/S) and the Norwegian Institute of Technology (NTH).

would like to express my gratitude to my supervisor, Professor Odd NI. Faltinsen for his

support and encouragement.

Many tanks are also extended to my friends at Marintek A/S, especially I.V. Aarsnes, H. Ruth and T. Boo.

Declaration

The model tests presented in Appendix A were performed together with J.V. Aarsnes and H.Rudi. The results in the main text is developed by the author.

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Abstract 2

Acknowledgement. 3

Nomenclature. 6

1 Introduction 9

2 Flow through and around nets. 16

2.1 Introduction to flow through and around nets. 16

2.2 Wake model. 17

2.2.1 Introduction to free turbulent wake flow. 20

2.2.2 Two dimensional wake flow. 24

2.2.3 Two-dimensional wake behind a single body in steady

flow. 25

2.2.4 Two dimensional plane wake. 28

2.25 Two-dimensional turbulent wake far downstream. 32

2.2.6 Plane wake in oscillatory flow 35

2.2.7 Vorticity transport. 42

2.2.8 Three dimensional axially symmetrical wake flow. 47 2.2.9 Three dimensional axially symmetrical finite velocity

defect. 55

2.2.10 Far field matching. 62

3 Uniform source distribution approach to flow through screens. 65

3.1 Flow through screens based upon G.I. Taylors approach. 65

4 Source model approach for flow through and around screens. 73

4.1 The mathematical model. 73

4.2 Source function 75

4.3 Numerical model 78

4.4 Numerical results 80

S Current force on fish farming structures. 85

5.1 Introduction to theoretical analysis of current forces on fish farming

structures. 85

5.2 Current force model. 86

5.2.1 Current force on a single net panel 88

5.2.2 Velocity reduction. 91

5.2.3 Current force on a system of cages. 93

.

,..

.... .

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6 Validation of current force model. 96

6.1 Uncertainty analysis of measurements. 96

6.2 Verification of the numerical model. 98

6.3 Uncertainty analysis of the numerical model 99

6.3.1 Uncertainty analysis of the drag force on a net panel. 99

6.3.2 Uncertainty analysis of the drag force on a cage system. 108

6.4 Comparison between numerical model and model test results. 117

6.5 Effect of neglecting structural compatibility. 123

7 Application of results 126

7.1 Direct application of the current force model. 126 7.2

Other application

of the

derived method for current force

calculations. 126

8 Conclusion. 130

A Model test. 132

A.1 Introduction to the model test. 132

A.2 Model test results. 132

A.2.1 Drag coefficient of a single net panel in current. 133

A.2.2 Single net panel in wave and current. 135

A.2.3 Cage system in current. 135

A.2.4 Velocity reduction. 144

A.23 Cage system in waves and current. 147

Referances. 148

...

. .

...

. . .

...

... .

.

...

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NOMENCLATURE

A - Area.

- Constant. - Matrix.

AF - Area of tube spanning out the net panel.

AN - Area of net panel normal tothe flow.

AP - Area of net panel parallel to the flow. - Characteristic width.

Constant

Ref/

- Function of stream function. - Constant.

Width of mixing zone.

- Constant

CD - Drag coefficient.

CDF - Drag coefficient of the tube spanning out the net panel.

CDN - Drag coefficient of the net panel normal to the flow.

C,P - Drag coefficient of the net panel parallel to the flow.

CL - Lift coefficient_ - Characteristic diameter. - Twine diameter. - Draft. dA - Infinitesimal area. dL - Infinitesimal length.

eu - Unit vector indirection of the flow. eT - Unit vector in directionof the net panel.

- Function.

F, - Drag force.

FDN - Drag force on net panel normal to the flow. FOP - Drag force on net panel parallel to the flow. FDF - Drag force on the tube spanning out the net cages.

FL - Lift force.

Force in x-direction.

Fyi Reaction force in node! in X-direction. Fyl Reaction force in node I in Y-direction.

Function. Gravity.

G1 - Inertial force on element I.

Wave height_

H(x) Heaviside function.

Imaginary unit.

Modified Bessel function of first kind and order i. J, Bessel function of first kind and order i.

KC Keulegan-Carpenter number.

K. Modified Bessel function of second kind and order i. Wave number.

L,I Characteristic length.

Mixing length.

1(z,t) Direction vector for lift force.

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-Tit - Length of element jl.

- Momentum.

Me - Moment about the pdint,cr., - Number of elements. - Newton.

n. - Normal vector.

n(z,t), - Direction vector for drag force.

Nc - Number of cages in direction of the flow. - Number of cages normal to the flow.

- Order of magnitude. - Pressure.

- Fourier transformation variable. - Source. strength.,

R,r - Radius.

- Polar coordinate. - Resistance.

Re - Reynolds number.

ER - Real part of a complex value..

r=u/I.1_ - Velocity reduction factor.

Re=DUN - Reynolds number.

S,Sc - Control surface.

- Body surface.

- Laplace transformation variable_ Sn=2D5.+1/2(D/1)2 - Solidity ratio.

St=f,D/U - Strouhal number. - Time

U,u - Fluid velocity in x-direction.,

11.11 - Normal velocity on the left hand side of the screen. Domain I.

Un - Normal velocity on the right hand side of the screen.. Domain B.

- Normal velocity of the control surface.

Uc - Current velocity.

- Body velocity in x-direction.

Use - Estimated body velocity in x-direction.

Ur - Radial velocity.

Uw - 'Oscillatory fluid motion. - Tangential velocity. - Free flow velocity.

V - Velocity vector.

- Tangential velocity on the left hand side of the screen. Domain I. - Tangential velocity on the right hand side of the screen. Domain II.

V. - Normal velocity through the control surface. - Body velocity in z-direction.

V,'

- Estimated body velocity in z-direction.. - Fluid velocity in y-direction_

- Wronskian.

- Complex velocity potential.

- Bessel function of second kind and order i.

X(i) - Vector of node positions in x-direction.

0

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x,y,z - Coordinates.

Vector of node positions in z-direction. z=x+iy - Complex coordinate.

- Angle 'between net normal and flow direction.. - Angle.

- Constant

- Small quantity.

8x Infinitesimal change in the variable x.

8(4

- Dirac delta function.

81 - Displacement thickness.

82 - Momentum thickness.:

- Phase angle.

et - Virtual kinematic viscosity. - Complex coordinate.

1(0

- Surface elevation.

- Transfer function for surge motion., - Transfer function for heave motion

0 - Angle.

Influence coefficient between drag force and parameter 1.

Or: - Influence coefficient betweendrag force and weight of sinker.

eel

- Influence coefficient betweendrag force and drag coefficient for the

tube spanning of the net.

- Influence coefficient between drag force and lift coefficient.

eel" - Influence coefficientbetween drag force and drag coefficient for the net panel normal to the flow.

0eel IP' - Influence coefficient between drag force and drag coefficient fertile net panel parallel to the flow.

ou Influence coefficient between drag force' and free flow velocity. - Influence coefficient between drag force and velocity reduction factor - Pressure drop coefficient,.

- Empirical constant. - Mesh size. - Characteristic length. Density of water. Kinematic viscosity.. di - Velocity potential. Error function. `P Stream function.

Turbulent shearing stress. Vorticity. Vorticity. a - Partial derivative. SCP) - Fourier transformation. Laplace transformation. Z(i) 113 1.) CD g(s)

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-1 INTRODUCTION

During the 1980'ies Norwegian Aquaculture Industry experienced an explosive growth. The

fish prices were high and the possibilities of profit were good_ For some years little or no attention was paid to the technical standard of the fish farms, mainly floating cage systems. Not until damages and collapses of fish farms led to escape of fish and thereby major

economic losses, the problem of environmental loads and structural strength were focused on. In the later years one has also focused on the environmental consequences of fish escape, due

to the risk of genetic pollution and spreading of diseases into the wild stock of species. It soon became evident that calculation of loads and response of fish farms were not such a straightforward task. For instance, there did not exist reliable methods for an apparently simple task as calculation of steady current forces on a net cage.

A large quantity of different structures used for floating fish farming exist, mostly made of steel or high-density polyethylene (HDPE). Figure 1 shows different concepts, the most common ones have been the types consisting of bridges with net cages between.

Fish fanning structures is a rather new type of marine structures from a engineering point of

view with special problems different from what we usually have in offshore and ship engineering. The floating part of the fish farming structures can be compliant and they can have very low buoyancy. The mooring system is also a significant part of the structure, and the restoring forces have to be almost horizontal at the connecting point due to the low

buoyancy of the structure. The net cages are non-solid and flexible. This imply that the flow

will partly go through and around the net cages. The net cages will be deflected due to the action of the current, and the form is given by equilibrium between the forces acting on the

net and the weights of the net and the sinkers used for spanning out the net cage. A principal

description of the flow through and around a cage system is shown in Figure 2.

The common fish farming plant is usually a long and slender structure. The plant is normally

situated with the largest dimension in the main wave and current direction. This orientation

gives rise to different types of interaction effects. First we can have cancellation effects in the total wave loads on the fish farm. Secondly we have important interaction between the flow

and the structure. This means for instance that the flow velocity in a given cage is reduced relative to the velocity in the upstream cages, due to the loss of momentum when the water

flow through a net. In addition the water may contain faeces and other types of pollution from

previous cages together with a lower contain of oxygen. Hence, the quality and the amount of water will decrease as the water flow through cages. The fluid interaction effect will depend on the solidity of the nets and the distance between the cages. Fouling of nets will

increase the problem of reduced water exchange.

Practical experience in fish farming has shown that sufficient water exchange inside the nets

is an important factor for the fish health and growth. It is an important factor for:

Available oxygen compared to the size and amount of fish.

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Mooring rope (8-2)i

Fish net (5-1) Anchor float (7-1) 'Mooring Picric (I 1-1)

The Bridgestone, "'Hi-Seas" fish cage.

The Parmocean. "semi-submersible" fish cage.

The Aqua System TOA fish farm.

Floattor frame pant (3-/) Cushion float (6-1)

Figure 1 Different cage .systems used in fish farming.

The Seacon fish *farming platform.

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From above

Side view

Figure 2 Description of the flow through and around a cage system. The flow will partly go through and around the cage system.

Necessary current velocity for exercise and well-being of the fish.

Although some research had been made for drag on single net panels, little or no attention had been paid to the analysis of current forces on net cages. The method recommended for

use by the authorities (DnV classification rules) was originally based upon results from wind engineering, Norwegian Standard Association: NS3479.

Early model tests, P. Werenskiold et. al. (1987), clearly indicated that the recommended approach for current force calculation was not applicable for the analysis of fish farming

structures. This is mainly due to fundamental differences in important flow parameters, such as shielding and deformation of the net cages. Actual calculations based upon the recommended approach showed that the method was not able to reproduce the main features

of the flow problem. For some cases results up to 3-4 times the measured force were found. On the other hand for certain conditions they were found to underestimate the current force.

Hence, it was concluded that there did not exist (or at least not known for us) any reliable

method for current force calculation on fish farming structures.

It was then decided (1987) to start a 3 years research program funded by the Royal

Norwegian Council for Scientific and Industrial Research (NTNF), Marintek AJS, NTH and

some of Norway's largest producers of fish farming equipment, aiming at a better

understanding of the flow through and the forces acting on a cage system exposed to current

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farming structures". The present dr.ing thesis is one part of this research program. The research program was divided into two' parts; model tests and theoretical analyses. The

theoretical achievements obtained in the research program is derived as a partof the present

dr.ing thesis. The results from the model tests and the current force calculation model are documented through a series of reports: H. Ruth et. al(1988a,1988b,1989,1990), G. Ltiland et. al. (1988,1989) and J.V. Aarsnes et. al. (1989a,1989b,1989c,1990a,1990b). A computer program for current force calculation is also prepared as a part of the project, this program is available for PC, see IV. Aarsnes eta (1989a,I989b).

Since we am working with a new type of marine structure, it was decided to perform

extensive model tests. Based upon the hypothesis that a net cage can be divided into a number

of net panels in the current force calculation, it was decided to perform model tests with a

Single net panel in order to establish relations between solidityof the net panel, angle, drag-and lift-coefficients. Extensive tests were also done with different cage systems, both flexible

and rigid net cages were used. The velocity distribution inside the rigid net cages was also measured. These tests, together with the tests with single net panels, were done in order to

validate and derive a theoretical method for current force calculations on fish farms. In the derivation of the model for current force calculation, it was natural to' start with the

analysis of the current force on a single net paneL This topic has earlier been investigated by

a number of authors, e.g Milne(1970) and Osawa et.al (1982). However, most of this work

has been related to the drag-coefficient on a rigid net panel as function of solidity, mesh type, Reynolds number and angle of attack.

The next step was to find the velocity distribution within the cage system, in order to find the forces acting on each net panel'. Our primary interest was to find a relation between the drag-coefficient for the net or the screen and the velocity behind it. The problem of flow through a two-dimensional screen has beeninvestigated by a number of authors L M Laws andJ.L

Livesey (1978) gave an overview of the work done inthis field. However, only a few have

investigated screens in infinite flow. Most people havebeen studying screens spanning over

the total cross area of a channel (wind tunnel).

0.1 Taylor (1944) was the first one to investigate the problem. He approximated the screen by a number of resistance elements (source elements). He assumed that each resistance element give rise to a flow outside its wake roughly equivalent to the flow produced by a source of equivalent resistance. In his analysis the drag coefficient for a flat plate will be

zero, which is clearly wrong.This result is related to a wrong reference velocity used in the

calculation of the force on a source element. If one uses the correct velocity in this type of

analysis one get C0=4.0 for a flat plate' in infinite flow. The correct one is C,,=2.0. This will

be further discussed in the main text

J.K. ICoo and D.F. James (1973) used a different approach. They replaced the screenby a distribution of sources and manipulated the stream function for this flow so that mass and

momentum balances across the screen are satisfied. Consequently this model predicts a flow

field which is realistic except for the discontinuity in velocity between the wake' and the external flow outside the wake.

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same linearized equations as used by Schlichting (1979) in analysis of wake flow. This means that we assume that the vorticity is convected downstream with the current velocity, and there is only diffusion in transverse direction. The screen is approximated by a number of cylinders with diameter equal to the twine diameter of the screen. Each cylinder will generate a two-dimensional wake. The wake behind the screen can be approximated by the

sum of the wakes from all the cylinders used in the approximation of the screen. This is due to linearities of the governing equations. We have also used the linear wake equations in the analysis of a plane two-dimensional wake, axially symmetric wake behind a single body and axially symmetrical plane wake. The solution of the non-steady wake equation is used in the

analysis of vorticity transport in the wake behind a body.

Based on the analysis of the flow through and around screens, we are able to estimate a relationship between the drag-coefficient and the ratio between the flow in the wake of the

screen and the free flow. This factor which is called "velocity reduction factor, r" is very important in the development of the method for current force calculation on fish farming

structures. The velocity in the near wake of the screen is almost constant in a plane transverse

to the flow direction. Further downstream the wake is mixed with the free flow outside the wake through diffusion, and the wake will vanish as we go further downstream.

In the development of the theoretical model for current forces calculations on fish farming structures one has benefitted from the physical insight obtained from the model tests. The

theoretical model takes into account the most important parameters such as drag-coefficient,

shielding between different parts of the fish farm and deformation of the nets. The derived method for current force calculation on fish farming structures can be described briefly as

follow:

The net cages are divided into a number of net panels parallel, normal or with

an arbitrary angle relative to the flow. For instance a square cage can be divided into four side panels and one bottom panel.

The fluid velocity is reduced when it flows though a net panel, in accordance with the velocity reduction factor.

Each net panel will experience a drag force and a deformation depending on the orientation, solidity of the net, weight of the sinkers and on the local free stream velocity. Due to the velocity reduction factor r. the local free

stream will differ along the cage system.

The total drag force is given by the sum of the drag force acting on each net

panel.

The main feature of the method is increased knowledge about the current forces acting on fish farms, hence the structure may be classified for a higher current velocity and higher waves. This may also reduce the dimension of mooring system components for a given set

of environmental conditions. Increased knowledge about the flow within and the deformation

of the net cages will also allow for improved design of the cage system to optimize the fish

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Figure 3 shows a comparison between model test, this method and the one recommended

before. We can see the proposed method reproduce the measurements, while the method based upon NS3479 clearly overestimates the drag force. In general, the derived method was found to reproduce the model test results within the rage of [0.9-1.2] times the measured drag force.

FD A NS3479 x Proposed Method

</.

x/---

Model tests

2

4

6 8 10 12 N

Figure 3 Comparison between the proposed model, model tests and results based upon

NS3479. N is numberofcages in the direction ofthe flow. F, is drag force on the cage

system.

Scale effects are considered as minor. We have used fullscale nets in the model tests. The drag force on a net is governed by the drag on each thread, and this parameter is correct

scaled.

The present dr.ing thesis will concentrate on the effect of current. However it is obvious that waves will have a significant effect both on the loads acting on the net and thedeformation

of the nets. The trend to move fish farms into less sheltered areas with significant wave heights of 5-6 m will increase the importance of reliable methods for calculation of wave effects. This mean snap loads in the nets, first order wave loads and mean wave loads. The snap loads can give rise to tensions in the net of 2-5 times that in current, while the mean wave loads can be of the same order of magnitude as that in steady current, this is highly

dependent on the wave height, wave periodand the motion of the top point of the net panel relative to the wave motion.

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A fairly simple method is proposed for the analysis of wave and current loads on a net panel,

as part of the research program, see G. Loland et.al. (1989). The model is based upon the

assumption that the fluid velocity due to wave and current can be superimposed. The analysis

shows that it is impossible to give an exact conclusion about the combined forces. They are strongly dependent on the geometrical shape of the net cages and of the floating part of the structure. In generally, wave effect will increase the loads on the structure. The wave loads are highly dependent on the structures ability to follow the waves.

The findings in this dr.ing thesis have improved the knowledge about the forces acting on floating fish farming structures. This knowledge can be used to improve the design of new

cage systems and operation of existing cage systems.

The thesis is divided into 8 chapters and 1 appendix. In chapter 2 we are dealing with the analysis of flow through screen based upon the approximation of the screen by a number of cylinders. We start with the derivation of the two-dimensional velocity field in the wake

behind a two-dimensional body, then we add the single wakes together to get the wake behind the screen. Then we extend the analysis to plane two-dimensional wakes, and merge these two

solutions with the single body solution in the far field. Then we solve the non-steady problem, which have relevance to oscillatory flow in waves, and also to the analysis of

vorticity in the wake. The next step is to extend the analysis from two-dimensional wakes to axially symmetrical wakes. We are then doing the same exercise with the axially symmetrical wakes as we did with the two-dimensional wakes. This chapter gives us a description of the

velocity in the wake behind a screen and the existence of the wake far downstream. In chapter 3 and 4 we investigate the flow through and around screens, based upon the approach outlined by (IL Taylor (1944) and J.K. Koo & D.F. James (1973), respectively.

From these chapters we end up with a expression for the relationship between drag-coefficient

and velocity reduction. In the end of Chapter 4 we study the influence of an inclination between the screen and the incident flow on the velocity reduction factor.

In chapter 5 we develop the method for current force calculation on fish farming structures in detail. We start with the analysis of a single net panel and extend it to a complete cage

system.

In Chapter 6 we validate the derived method, through error and sensitivity analysis together with comparison with model tests.

In chapter 7 we discuss some other applications of the model for current force calculations

Chapter 8 contains summary and conclusions.

Appendix 1 gives an introduction to the model tests, together with some of the main results from the model tests used in the thesis.

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2 FLOW THROUGH AND AROUND NETS.

2.1 INTRODUCTION TO FLOW THROUGH AND AROUND NETS.

Two different model tests were performed in 1988 and 1989 as a part of the research program "Current forces on and flow through fish farming structures". Both these model tests included

investigation of the flow through and around nets. The model tests are briefly described in

Appendix A.

As a result of the model test in 1988 a method for calculation of current forces on and flow through a conventional fish farming structure was developed. The method is described in detail Chapter 5. One important parameter in the method is the velocity reduction factor r, define as the ratio between the velocity in the wake of a screen or a net and the free flow

velocity. The drag force on a net panel is a function of the velocity. Hence, to get an accurate

estimation of the drag force one must know the incident velocity. This is not a problem if there is only one screen, but the problem rises if two or more screens are placed after each other in a row. So the main question is; What is the incident velocity at a net panel placed behind a given number of other nets?

Before we are able to find the velocity distribution behind a series of nets, we must first find

the velocity distribution behind a single net panel. This chapter together with the next two

chapters describes the calculation of flow through and around screens, with special emphasize

on the velocity reduction factor r , and on the relation between velocity reduction and drag

coefficient. The theoretical calculated velocity profilebehind screens is also compared with measurements. The analysis are based upon four different approaches. Three of this models

are based upon two-dimensional assumptions while the last one is based upon an axially

symmetrical approach.

The screen can be divided into several circular cylinders, with diameter equal

to the twine diameter for the net. Each cylinder will generate a two-dimensional wake, which can be calculated from the linearized

two-dimensional wake equations. These equations are linear, which imply that the

total wake can be approximated by the sum of the wake from each screen element. The most important parameters in this approach are the

drag-coefficient and the eddy viscosity.

The screen can be approximated by a number of axially symmetrical bodies.

Each body will generate a three-dimensional axially symmetrical wake, which can be calculated from the linearized axially symmetrical wake equations.The total wake can then be approximated by the sumof the wakes from each sub element. The main parameters in this approach are the drag-coefficient and the eddy viscosity.

The screen can be considered as a uniform distribution of "centers of resistance". Each element of resistance produce a wake. The flow on the outside of this wake is roughly equivalent to the flow produced by a source

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with the same resistance. The main parameter in this approach is the pressure drop coefficient_

4. The screen can be looked upon as a distribution of sources, as described by

J.K. Koo and D.F. James (1973). Using stream function theory, one can calculate the velocity profile outside and inside the wake behind the screen.

The most important parameter in this approach is the pressure drop coefficient.

This chapter describes the flow in the wake behind a screen based upon the wake equation approach. First we investigated two-dimensional flow, then we look at axially symmetrical flow. The two source flow approaches are described in the next two chapters.

2.2 WAKE MODEL.

Our main object is to find a relation between the velocity reduction for flow through and the

drag force on a net. One way of doing so is to use a model we called "Wake model". The principle of this model is to describe a global wake by a distribution of wakes behind sub elements. These wakes can either be two-dimensional wakes behind cylinders or

three-dimensional axially symmetrical wakes behind axially symmetrical bodies.

Before going into details of the analysis, some introductional results are shown. A screen can

be divided into several cylinders as shown in Figure 4, with diameter equal to the twine diameter and spacing equal to the mesh size.

The two-dimensional velocity profile in a turbulent wake behind a single cylinder can be

calculated from the linearized turbulent wake equations. The solution is given by Schlichting (1979), and the derivation is shown in Chapter 2.2.3.

u(x,Y) = 1.0 - 0.95. CD-Di .exp[- Y2

U_ x 0.0888-CDDx

CD is the drag coefficient and D is the diameter of the cylinder respectively. icy is the coordinated of a point in the wake. The above result is based upon a linear differential equation, thus we can add the contribution to the wake from all segments of the screen. We will neglect the hydrodynamic interaction between the cylinders at the inflow. This is a reasonable assumption if VD > 5-6, (X is the mesh size). This is the case in our problem, except near the knots where the cylinders cross each other. The interaction at the knots will to a certain extent be taken into account through the double artificial cylinder at that point. From the model test see Appendix A, we have estimated drag-coefficients for the total screen, not the drag-coefficient for each thread in the screen. So in order to have a consistent solution for the current force, each thread must have the following equivalent drag coefficient:

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A

_

Figure 4 Cylinders used as an approximationofa screen.

FE"" = E:"0 FL,"

71p

UA=

p

C= CIA""

-

°

- Total number of threads. A - Area ofthe net panel. Di - Diameter of thread no=i.

- Lengthof thread no=-i.

(2)

D.

Mesh size

1

z.

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Based upon equation (1), (2) and Figure 4 we can derive the following expression for the

velocity profile of the wake behind a screen.

1

D.

CY-y)2

u(x,y,z) = 1.0 - 0.95-EN

C/D

'

j

expfr i

U_

'''

x-x, 0.0888 C D.i'D j(X -xi) iip)

CD/Di

(z-z7

-,!0.951N1

0.0888 4,

Di(x-xj.)-N1 - Number of threads in y-direction.

- Number of threads in z-direction.

CD j - Drag-coefficient for threadno=i.

x,YA - Field point for calculation of velocity. - Source point of screen element.

Calculation for an actual screen shows that the velocity in the wake is almost constant except

at the edges of the wake, where there are a continuous change into the free flow: Table 1 below shows a comparison between measured velocity in the center of cage 1 in the model

test, see (Appendix A) and the calculated velocity from this method. The result is for a net,

with drag coefficient Crra=0.326, which gives an equivalent drag-coefficient for each thread

of Cou'd=1.37. The drag coefficient for each thread is somewhat high compared with

measurements of drag coefficients on smooth cylinders for the actual Reynolds number.This

imply that there are some interaction effects between the different threads. The table shows that the agreement is good, so we may conclude that this simple model gives a good description of the velocity reduction factor for the flow behind a normal screen in the near field. We will later in this chapter show that the wake calculated based upon the above

approach exist too far downstream.

Table 1

Calculated and measured velocity in the wake behind a screen of solidity Sn=0243, velocity U=0316 mls and Re=570.

Measured value Calculated value Drag-coefficient i CD '=O.32&

C'= 1.37

,Velocity reduction factor, r. 1=0.85 1 r4k84 i x-xi

(21)

We will now go into details of the solution, to get a more accurate solution in the whole

fluid domain. The following analysis will contain the above solution, the solution for a plane

wake and merging of the above solution with a far field solution. It will also contain the

solution for axially symmetrical flow and oscillatory plane flow.

2.2.1 Introduction to free turbulent wake flow.

When a body moves in a fluid at rest or placed in a stream of fluid a wake is formed behind

it, see Figure 5 . The velocities in the wake are smaller then those in the main stream, and the losses in velocity in the wake amount to a loss of momentum which is due

to the drag on the body generating the wake. The width of the wake increases with distance

from the body, and the dip in the velocity profile gradually levels off.

free flow

wake

>-X

Figure 5 Wake behind a body in a uniform flow. U is the free flow velocity, u is the velocity in the wake and u,=U-u. The pressure is assumed to be constant over the width

of the wake.

Qualitatively such flows resemble similar flow in the laminar region, but there are large quantitative differences due to the far larger turbulent stresses. In the following analysis

laminar flow is used as a basic solution, then these solutions are modified to take into account the turbulence through an artificial eddy viscosity.

(22)

Free turbulent wake flow is of a boundary layer nature, meaning that the region of space in

which a solution is being sought doesn't extend far in transverse direction compared with the

main flow direction. Thus, the transverse gradients are much larger than the longitudinal gradients. Consequently, it is permissible to study such problems with similar equations as

used in boundary layer flow.

au au au

lap

a21.4

-F 4.

"TX +

117 -

17TX ay2

au +

au± vat!

lap

1 at

aT u-aT + ax p

au av

4.

0

ax ay

Here 't denotes the turbulent shearing stress. The velocities u and v are the real velocities in the laminar flow case, while they are the mean velocities over a sufficient time in the

turbulent flow case. The derivation of the turbulent boundary layer equation can be found in different text books, e.g see Schlichting (1979).

To solve the system of equation for turbulent flow, it is necessary to express the turbulent shearing stress by the parameters of the main flow. This elimination can only be achieved

through a semi-empirical assumption. In this connection it is possible to make use of Prandtl's mixing length theory.

au au

4

t

2.17 I ay

where the mixing length I is to be regarded as a local function. Or, it is possible to use

Prandtl's virtual kinematic viscosity hypothesis.

au au

= p = p

- u

(6)

ay mua ay

where b denotes the width of the mixing zone and IC is an empirical constant.

= - um.) (7)

at is the virtual kinematic viscosity, which can be assumed constant over the whole width and, hence independent of y. It is the last assumption which will be used in the further

analysis.

Before we continue with the solution, we will make some assumption about order of

magnitudes. In this way we can find the type of law which governs the increase in the width of the mixing zone and the decrease of the velocity reduction profile with increasing distance.

The analysis here is to a large extent based on Schlichting (1979) and it is included for

explanation purpose.

It is usually assumed that the mixing length I in turbulent jets and wakes is proportional to the width of the wake, b.

(5)

laminar flow

turbulent flow (4) +

(23)

1

= [3 = constant (8)

And that the rate of increase of the width, b, of the mixing zonewith time is proportional to

the transverse fluctuating velocity.

Dbau

. v' « 1._ .. 1-

ulams

Dr ay b

Db 1

_ = co

.U = constarzt13-u,

Dr b

l'u

Where the left hand side is the substantive derivative:

D_ a ,,a

Dr ax

7

Which for the wake in Figure 5 when the velocity defect is small, take the form:

Db ab

=u_+v

ab db

U (11)

Dr "ax dx

since b is a constant in transverse direction. Equating the two expressions for the substantive

derivative of b we get.

db

= constant13-u o 13-u1 "dx

dl, u,

Equation (12) holds for two-dimensional andcircular wakes. We can then use conservation of momentum to get an expression for the drag force acting on the body generating the wake.

Fd=

71p-CD-Ud"-rc

=su-(U -

u)ds

= U - u

A u, < U_

1= r_Lds

2 D

UU_

u, CD.c1"

U_ b"

Here d is the characteristic diameter of the body generating the wake, n=1 for

two-dimensional wake and n=2 for circular wake.Equation (12) and (13) gives us an expression

for the width of the wake.

=

=

(24)

1151 13 db U, CD-d" clx U_ b" b "-db f3CD4"-dx T141 .13,C0-cr% n+1 b cc (pCifd"-x)'

This states, that the width of the wake increases proportional to the squareof the distance in two-dimensional flow and proportional to the cube root of distance in circularwakes. From equation (13) and (14) we can get an expression for the decrease in the velocity defect.

u,

Col"

U_ b^ u, CD-cl" U_ (pCD-d'i-x)ma'). U_ (CDcl")1/(n.1) 13"x"

Hence the velocity defect decreases proportional to the square root of distance in two-dimensional wakes and proportional to the distance in 2/3 power in circularwakes.

From equation (7) we have that the virtual kinematic viscosity, called eddy viscosity, is proportional to the width of the wake and to the maximum velocity defect. Hence, we have

that: et, k-bitixan C cc 1C(pcp.d..x)0.1),u...( D )110.41 Ps'X' I I-.

S

cc U {CD2-62"Fr% (16)

Et cc U_CD"d two-dimensional wake

24

u

Et cc u_C.C17% circular wake

This implies that the virtual kinematic viscosity is independent of 'distance for

two-dimensional wakes, and that it is inverse proportional to the cube root of distance for circular wakes.

u

(25)

2.2.2 TWO DIMENSIONAL WAKE FLOW.

The flow in the wake of a two dimensional body (see Figure5 ) is of a boundary

layer type. This implies that its growth is governed by the two-dimensional boundary layer

equations (17).

In the following, we use the solution for the laminar flow as a basic solution of the problem.

Then these results are modified into turbulent flow. This is a very simple task in two-dimensional flow since the only difference is to use eddy viscosity instead of the kinematic viscosity, and that the velocities u and v arethe mean velocities. Hence, in two-dimensional

wake flow it is exactly the same set of equation that governs the growth of the wake.

au av

=0

ax

Du au Du 1 Dp

??u+

u_ + v__ = -

+ V Laminarflow

IV)

ar ax ay

777

ay'

au

+ u v_

au

auap

+ c a2u Turbulent flow

at ax ay p ax tay2

Assuming the pressure to be constant in the transverse direction over the wake, using Bernoulli's equation on the outside of the wake we get:

au au au au_() a2u (18)

at dx dy

We will approximate the velocity in x-direction by the free flow and a perturbation velocity:

u(x,y,r)=U_(t)-tii(x,y,t) (19)

This means:

a(U_-14,) D(U_-u1) -u1) DE I _(t) a2(U_-u1)

+v

at ax

= +V

(20)

Dui Dui au,

at ax ay ay2

From continuity and the boundary layer assumption about order of magnitude, i.e that the

derivatives in longitudinal direction is much less then the derivatives in transverse direction,

and that the perturbation is velocity small compared with the free flow, we get:(8 is used

as index for a small quantity)

U_(t) u1(x,y,t)=0(5) U_(t)=0(1) (21)

au av

au -u avav

+_.0

v(x,y,t)=0(8)

ax ay --";7"-- ay ax

ay2

(26)

The governing equations in terms of order of magnitude is given as: au, au,

au, ,,au,

at ax ax ay ay'

0(5) + 0(5) -0(52)+0(52) = 0(5)

Neglecting terms of order 0(52) we get the following linear differential equation for the velocity defect when the pressure is constant over the wake:

au au a2u

=V I Laminar flow

at - ax ay 2

au au Yu (23)

I +U 1=e 1 Turbulent flow

at - ax ay2 u(x,y)=0

For the time independent case the governing equations takes the form.

au a2u U =V Laminar flow ax Du, D2u U =e ax

lirny u(x,y)=0

The wake solution of equation (24) for a single body is well known, e.g see Schlichting (1979). In the next sections the solution for the following cases are given: The wake behind

a single two-dimensional body in steady flow. A plane two-dimensional wake in steady flow

and in oscillatory flow. The wake behind a axially symmetrical body and a plane axially

symmetrical wake in steady flow.

2.23 TWO-DIMENSIONAL WAKE BEHIND A SINGLE BODY IN STEADY FLOW.

The two-dimensional wake behind a single body is the principal solution in the wake model.

It is therefore natural to start the analysis with this solution, although the final solution is given by Schlichting (1979). The solution is equivalent for laminar and turbulent flow. The only difference is that the eddy viscosity is used in turbulent flow and the kinematic

viscosity in laminar flow, see equation (24).

Before we can find the solution we must have an expression for the initial condition. For the

wake behind a single cylinder in steady flow the initial condition take form of a Dirac delta

function:

ui(x=0,y)=A15(y) (25)

because the velocity defect get larger and more narrow as we approaches the generating point

at x=0, the constant A can be found from conservation of momentum. This is given as:

(22)

(24) Turbulent flow

-u

(27)

F p=pU_I-uldy =

pU4

AIS(y).dy = p C D-D -U! 2 1 141(x =0,y) = CD -D _-8(y) 2 1 A = D 2

The governing equation can be solved by first taking the Fourier transform in y-direction:

KP) = 7(11) = PO') vxP(- iPY)dY

When gp) is known, we can find f(y) by the inverseFourier transform.

f(y) = f.:Kp)-exp(ipy).dp

27t

Taking the Fourier transform on both sides and using the far field boundary condition we get.

+vp2ii =0 (29)

ax

u

This first order differential equation has solution.

=C-exp( - vP2T) (30)

U

The constant C is given from the Fouriertransform of the starting condition at x=0.

T 4,= C vir4z) (31)

The real solution is now given by an inverse Fourier transform .

ul(x,y)=-471 C D-D -U_Texp(42-.x)-exp(ipy)dp

u1(x,y)=_L c exp (

2x) ico

spy + isinpyldp (32)

47c

1 p2

is,(x,y)=C D-D

exp(-v r).cospydp

47: U

This integral is given in Gradshteyn & Ryzhik (1980) on page 480.

1 It U y2-U (33)

u (x,y)= ___C 0-130-U Laminar flow

47t 4v -x

This is the same solution as given by Schlichting for laminar and turbulent flow. Thesolution

for turbulent flow is expressed directly by changing the kinematic viscosity to the eddy

viscosity, v-9e.

(26)

(28)

1 It U y2-U (34)

ui(x,y)=_CDD-U:

- txp( -_ ') Turbulent flow

4it . Et 'X 4E, -x

The eddy viscosity is still unknown, but from measurements by Schlichting (1979) we have that:

=0.0222-U_C0 -JD (35)

The final solution for the velocity profile in a turbulent wake behind a single cylinder in

steady flow is given as:

n 2

14,(x,y)=0.95-U_(CD-)112.exp(- Y ) (36)

x 0.0888-CD-. D .x

This is the equation we used in the beginning of this chapter, equation (3).

Equation (36) can also be used in a similar manner to calculate the two-dimensional wake

behind a two-dimensional net, if we approximate the net by a row of cylinders, see Figure 6. u(x,y)

= 1.0 - 0.95r'

\I 'exPH (Y-Yi)2 (37)

X -X. 0.0888CD;D:(x-xj)]

- Number of threads in y-direction. Drag-coefficient for thread no=i.

D, - Diameter of thread no=i.

x,y - Field point for calculation of velocity.

xi,Y; - Source point of screen element.

The velocity reduction factor r behind a two-dimensional net with drag coefficient CD"'=0326 , twine diameter D=1.83 mm and mesh size X=15.5 mm, which is the same net as we have used before, is r=0.84. The velocity reduction is calculated in a point one

length dimension behind the net, with one length dimension we mean the transverse width of

the screen. The equivalent drag coefficient for each cylinder is now CDth'=2.74, since we in this case have half the cylinders available to approximate the screen with compared to

the previous case. This two-dimensional distribution of cylinders will be used in comparison

with the plane wake model, but in the comparison case we will use a screen with drag coefficient CD=0.2, twine diameter D=0.0033 m and mesh size X=0.02 m, which gives an

equivalent drag coefficient of CD=1.2 for the threads. CD'D'

(29)

1.3

Figure 6 The net can be approximated by a two-dimensional row of cylinders. The total

wake is given by the sum of the individual wakesfrom each cylinder.

2.2.4 TWO DIMENSIONAL PLANE WAKE.

In the previous subsection we found that the wake behind a screen or a net is almost constant

in transverse direction over the width of the screen. This imply that the wake flow is very

similar to a velocity defect shown in Figure 7. The main difference between the real wake and

the mathematical model wake, called a plane wake, is that the real wake has continuous velocity gradients while the model wake has an infinite velocity gradient in transverse direction at x=0, see Figure 7.

The velocity defect in the wake is small compared with the free flow velocity. This is basically same problem as we solved for the single body case, but with another initial condition for x=.0. In this case the initial condition for x=0 is

(x = 0,y) =a.b-t = a-H(LI2 - iyi).U..

b=-1 for ly l<1.12 (38)

b=0 for ly I>L/2

Where H is the Heaviside function, with value 1 for positive arguments and 0 for negative arguments. The constant a is related to the drag on the body generating the wake, and it is

given as:

X

(30)

y

U (x=0, y)

Figure 7 A two-dimensional finite plane wake can be used as an approximation of the flow through a screen. U is the free flow and u is the velocity in the wake.

1

a = 1 (39)

FD = p-U -

I-

u

_p-C

CD

-

2 2

This imply that the velocity reduction factor is given as r=1-1/2CD, which is a quite good

approximation as shown later.

The Fourier transform in y-direction of the initial condition for x=0 is given as.

1711(x=0,P)= a.b.U.:exp( -ipy)dy=a-U_1-sin(PL12) (40)

The solution for the Fourier transformed velocity is then given as.

sin(pL/2) -____x)-__xvp2)

a-U_-2- (41)

The real solution is given by the inverse Fourier transform.

ui(x,p) = a-U -2-sin(pL/2)txp(--xvp2)-exp(ipy)dp (42)

27t p

The solution of this integral can be found after some pages calculations. Thus, we get the

following expression for the development in space of the plane wake.

ill (x, y)=U u (x, y) L/2 -L/2 = =

(31)

le La +y

La-y

-

)+0(

2

2Fri

2 vxn/r../..

CD-U 112+y L/2-y

-ie.(

)40(

4 2 vxr/FL 2

vxr/r)1

/.. Icb is the error or probability function given as ia

4)(x)+-2

41)3dz' 1(44)

v it the kinematic viscosity for laminar flow, for turbulent flow v is changed to the eddy viscosity c, which is unknown. The best we can do is to usethe value given by Schlichting (1979) for a single body. The solution for the wake behind a single body is a special case of the above equation, which we can get if we let D=L and then take the asymptotic expansion for large distances from the screen. The Taylor expansion of the error function about y is given as

2F/U_

2 writ-T.

?Far

"YAI "

L/2 +Y Y .a(1)(z)ii +1(

La

ab

2 2)Ivx1U az2 I

2vx/u

+0((

La

)3) (45) L/2-Y

).-0(

Y ),I. 1.12 .3(1) WI

0(

2 vxr/r/.. 2,rvxffi_

2F/T.r..

az z-zlit'amr: i(46)

-1(

112 )z a1/41)

+((

L

,2 '

I

v, w(i2

)3)

2

2 vxnr_

LIZ

earu-

2 vxsr/r/...

Equation (43),(45) and (46) gives the following asymptotic expansion for the plane wake, i.e when the transverse dimension is small compared to the distance from the generating point.

C U

u

!(47)

4 v x-x 4v x

This is the same solution as the one originally given for the single body case, which shows that the single body wake is a special case of the moregeneral plane wake.

Based upon equation (43) we can get an expression for the mean velocity over a prescribed

domain in transverse direction, i.e the mean velocity in the area between a and b in transverse

direction. This velocity is of interest in the current force calculation model on fish farms described in Chapter ui(x,y) (43)

)_0(

LI2 y 5.

(32)

uzo,

4

-= V1 24 41(

11211 )+JLe

2(a-b) 2 vxsii/:, 2ilvx1U_

Fr

2177177

4 112-y ),

L12-y .0()_ 1

ZrWortr

21/v.

.0

2Fx-/r/..

Fr

Figure 8 shows the mean velocity as function of distance from the screen for three different

positions in transverse direction; One behind the screen, one half covered by the screen and

one outside the screen. We see that the mean velocity depends on the distance both in longitudinal and transverse direction and on the initial velocity defect We can see that the

velocity defect is reduced to the half about 25 length dimensions of the screen downstream. In general, we see that the wake exist far downstream.

Mean veloctiy:

Behoo

----

HoLF behold!

I ---

Outs de

=[0(

fb

40-a)

20/TVT)+0(

-2t; 0.6 0.4 0,.2 0.0 Li2 Y )1 dy r(48) l 1 I 1 r /

..-/

So 'IOU ISO 200L

Figure 8 Mean velocity in the wake behind a screen as function of transverse position and distance from the screen. C0=0.4, L =1.0'm and U=1.0 mls.

to

0.8 CD .ai(x,a<y<b) 2 CD x/L CD

(33)

2.2.5 TWO-DIMENSIONAL TURBULENT WAKE FAR DOWN STREAM.

Both the plane wake model and the sum of cylinders model gives a good description of the velocity defect in the near wake, i.e at distances less then one length dimension of the screen

or the non-solid body generating the wake.

The next question is: How does the wake act far downstream. From the general wake theory we know that the form of a turbulent wake far down stream is independent of the local effects

at the generation point. It is only affected by the drag force on the body and of the eddy

viscosity.

Hence, the two-dimensional wake behind a screen far downstream is also given as:

CD

(x,y) =0.95 )1a zxp(- ) (49)

0.0888C,D-x

Here CD is the drag coefficient and D is the transverse dimension of the screen respectively.

Figure 9 shows a comparison between estimatedvelocity defect behind a two-dimensional

screen calculated by different models, a plane wake model, a sum of cylinder model and a

single cylinder model. The geometrical properties of the two-dimensional screen is shown

in Figure 6, mesh size X=0.02 m, twine diameterDz--0.0033 m, C0"=0.2 and CDth'=1.2. We can see from Figure 9 that the sum of cylinders model gives a wake which exists too far down stream compared with the single body wake which is assumed to be asymptotic correct far downstream. While the plane wake model gives a correct asymptotic value, both far down

stream and at the screen itself. This correct asymptotic behavior is also shown in equation

(47)

The single body model which is assumed to be asymptotically correct far down stream, gives

completely wrong velocities near the screen. This is also to be expected since the boundary

layer equations leading to the expression is only valid further downstream then 50-100 dines'

Cole, see F.M. White (1974).

The reason why the sum of cylinders model gives wrong values far down stream is the eddy

viscosity which is used. The eddy viscosity is related to the width of the wake. So the question is: What is the width of the wake when wakes from different bodies merges? In the above calculations we have used an empirical value for eddy viscosity given by

Schlichting (1979)

=0.0222 -C,LI -U. (50)

Which in the sum of cylinders model becomes.

=0.0222 CD:D:U_ (51)

To get a better solution far down stream we have to modify the expression for the eddy viscosity in such way that it takes into account for the merging of the wakes from different

bodies. The following functional relationship between eddy viscosity and distance is used:

(34)

0.110 Dr.05 M.00 0.29 0.10 .0.os 0.00r 0 x/cd L 500:0 21 10.20 vitri 0.15 0.10

'MOd f ired .sum of cyll

ti

1

Single body model Sum of cylinder mod

Plane wake 0.05 0:00

x/cd L

50.0

x/cd L = d00.0'

0.20 d0201 U 1 0.15 4).5 0.10 0.05 101.00 1 x/cd L = 1000.01 0.20 2

Figure 9 Velocity distribution in the wake behind a screen as function ofdistance downstream. Results from four different modelsare shown; a single body model, a

of ncer ul Oh M. 0 ;05 -r0.00 0

x/cd L = 1.0

x/cd L

10.0 U1 0.20 0.15-0.10 0.05 -0.00 0.15 -

-0 =

-0 2 3

-0 3 0.15 -3 3 2

(35)

a x

c, +a_a7(x-tanh( :.5)))

D (52)

=e, Urn Et r-E0(1 +a)

This function is used due to its behavior. It has the asymptotic value 1 for large arguments

and it vanishes when the argument approach zero.It is also easy to integrate, in that way we are able to get an analytical solution.The value of the constant a is chosen in order to merge

the eddy viscosity for a single body far down stream.

E0-(1 +a) =0.0222CD L-U_

0.0222CO3-(2,-U_-(1+a)=0.0222C0EI 11 C D -1 Co,q, 1i a =N -1

Where N is the number of cylinders used in theapproximation of the screen, CD is the drag coefficient for the screen and CD,; is the drag coefficient for cylinder no=i. L and I is the length of the screen and the diameter of cylinder no=i respectively. The governing wake

equation is now given as:

au, a2u U ° 1+(N-1),..._a(x -tanh( ))] ax ax CD-d) ay2 11

-1)

a

(x-tanh())]-/ =0

7

Cif(' 11 (54) vo-x(1+(N-1)-tanh( cx.d) 1222_°_. 1 N

-57= u_

+( C D-D 1 ic -U_ y2U u, =C ,.,-D-U- - ) -exp(- ]

' 47t - \ va -x-( 1 +(N - 1)-tanh(xICD-ci) 4v x -(1 +(N- 1 )-tanh(x/CD -d)

We can now modify the expression in the sum of cylinder model, to get a better description

far down stream using equation (54) instead of equation (36).

The results from this approach is alsoincluded in Figure 9. It gives almost exactlythe same

results as the plane wake model.

Based upon the above calculations, we concluded that the sum of cylinder model

over-estimates the velocity defect far down streamif the eddy viscosity is related to theindividual thread, without taking into account the merging between the wakes in the eddy viscosity. On

U_

ii

(53)

(36)

the other hand, it gives very good agreement, if we use an eddy viscosity that take into

account the merging between different wakes. It gives almost exactly the same results as the plane wake model.

Hence, both the plane wake model and the modified sum of cylinder model gives a good description of the development of a 'plane' two-dimensional wake. These two models give

asymptotically correct expressions far from and very near the screen. It is not possible to give an exact conclusion on the behavior between these two limits, due to the lack of

information. However, it is plausible that these two models, which gives correct asymptotic behavior, also describes the intermediate region well.

2.2.6 PLANE WAKE IN OSCILLATORY FLOW.

Oscillatory wake flow is of interest in a variety of problems. It is important for all types of

structures placed in the wake of other bodies or effected by its own wake, i.e wave loads on nets, risers and jackets.

We will in this subsection discuss some aspects about oscillatory wake flow, with special emphasize on the solution of the governing differential equation. We will derive a solution of the non-steady wake equation. The same initial condition as for the plane wake case is used, this imply that we are working with a constant velocity defect over a given area in

transverse direction. The development of a wake in oscillatory flow is governed by the

non-steady differential equation:

Du au a2u

--1 4-U '

at as

ayr

This equation can be solved by first taking the Fourier transform in x-direction. This has to be done carefully since u, is discontinuous at x=0.

-1.41+_=-U (x=0-,y,t)-u1(x=0',y,t)]

ay2 at

-The right hand side is the initial condition at x=0, which is given as a drop in velocity across the body generating the wake at x=0.

U_(r) =U. + UB-cos(cut+e) (57)

Thus the Fourier transform in x-direction of the wake equation takes the form:

-v 141(x=0,y,t) (58)

ay - Dt

+i p

(37)

-The next problem is to find the initial condition for x=0. This is the main problem, due to the

different behavior of the initial condition when the flow returns over the body and when it does not. So it is convenient to divide theanalysis into two different problems: One for the case when the oscillatory flow is

larger then the steady current and one for

the complementary case.

We will use a drop in the velocity as initial condition at x=0, i.e a uniform reduction of the incident flow over the width of the body generating the wake. Which is the sameinitial condition as used in the steady state case.

(62) U Jx,y,t=0)=0 2 We also define:

.40.

6

U +UB -cos(o) t +e)C

for LI,U,

Uclanh(k1)+UR-cos(un+it/2) for U B<U c

[

(63) Uct+____-sin(c)t+E) for U g(t)-= g (t)dt= 0) U (64) c

_.1n(cosh(k-t))+_-sin(cot+n/2) for UB<Uc This equation can then be solved by taking the Fouriertransform in y-direction.

+ (vs 2 +ipU ji;=U:ru,(x=0,y,t).exp( -isy)dy (59) where the right hand side is the Fourier transform in y-direction of the initial condition. The

free flow U_ is composed by a steady current U, and an oscillatory part Us(t).

U_(t) = U c + U8cos(o.u+e) (60)

We will assume that the wake generation starts from rest. This is not a necessary condition,

but we will use it in the present analysis.

U_(t,y,r=0) = 0

ii

U (61)

E =arccos(

This equation has only solution for U, Ub. If the oscillatory motion is less than the steady current, we have to introduce a time dependent current to start up the solution from rest in a smooth way. This can be done by defining the free flow as :

(38)

The constant a relates the velocity drop to the drag force on the body. It can not be related

directly to CD, as a=1/2CD as in the steady state case, since we have a time dependent problem,

which gives us a change in momentum in the control volume. However, for our physical problem resulting in this analysis, we have that the Keulegen Carpenter number of the individual threads controlling the drag force are large, hence Cd is assumed to be independent of the time t. We further assume that we can neglect the effect of previous

generated wakes on the initial condition.

The Fourier transform of the initial condition in y-direction is given as:

-u-i(x=0,s,t) = a.U2(t)-2-sin(sU2) (66) This equation gives us the following differential equation for the Fourier transform in x- and y-direction of the velocity defect.

at +(vs2+ipUJ17;=a-U_2-2- sin(sL12)

(67)

The solution of this first order differential equation is given as.

7=e J2 U2

s112)

'Pu-)aidti+C) (68)

From the initial condition at t=0 we have that.

ul(x,y,t=0)=0 72;(p,s,t=0)=0 C=0 (69)

We have here assumed that the flow starts from complete rest, this is not a necessary condition, it is sufficient to assume that th(x,y,t) vanish for t=0. Thus, we can write the

solution as:

LT=e-"'-e-P20)-12a(g1(t)

sin(sL/2).)2--u7= It2a(g/(0)z.sin(sL/2) ".°(`-`/-6. -'°<8(1)-g"dr

The inverse Fourier transform in y-direction is given as: .1-177-e"Yds

2rc

Interchanging the order of integration.

-

1

It

, 2 sin(sL/2) _

u 2a(g (r)) (72)

s

The last integral is the same as the one solved in the steady state case, equation (42) . Thus the Fourier transform in x-direction of the velocity defect is given as.

u1(x=0,y,t)=a-Q(t)b where

b.

= (t)+I(L12-I ll<L/21 [ for y LO for ly1>L/2 (65) I) .e u

(39)

fcgto)2?-ipurco-ett[oc

La+y

)+0(

-y ndt,,

2v(t-7)

2 '1,V77")

The final solution is then given by the inverse Fouriertransform in x-direction. u r"; -e"`tdp

I 27c Interchanging order of integration

,4,))210( L/2. 4.y )4.0( L/2-y

-Li Le.F04,0)-gnmdpi.de,

2r/(7)

2 1,177)

The last integral has to be investigated more carefully. This integral is of the same type as a delta function. Equation (76) shows that it is a delta function:

.1 1.-e-4gsc1)-rwa.eipsdp = sr'(e-iKeit-i(r') 4 27t

IL

50.e

-440-e6)=1-fre -indx

f = 8(z -(8(0 -g(e))) = oiy

However, the integral is a delta function relative to the variable x while the next integration is with respect tot; so we must change the integration variable from x to

u=x-(1)-et I))

du

=et') .<=> du

di' g'0') (77)

u,i

a= (g (2

t. )) tout L12+y

L/2-y

)+(pc, [u=(x-(g(1)-g(e)))1 du

2 g 101

2 NiT'7)

This integral may be. a bit strange, but it can beinterpreted in the following way

The delta function has the property that it picks out the function value for that t' which sets

the argument in the delta function equal to zero. If the argument in the delta function has no zero points, then the velocity defect is zero for that combination of x and t.If it has solutions,

then the number of solutions can vary from one to infinity, depending on the ratio between

ILIc and Un. In general, we can say that the delta function describes how many times one part

of the wake has passed a stationary point x. If there is no solutions, then the wake hasn't

reached that point yet. If there is one solution, than the given point has been passed once,and so one. If 1.1c>11.1, then there is one or none solutions. If Uc<Us then there can be more than

one solution.

rap

r(741 (75) t'. (t-rt) (76)

(40)

The final solution for the plane oscillatory wake is given as.

Wet@ -rdi

N is the number of zero crossings of the corresponding zero point.

L12+y ul(x,y,t) =

2Ft7

ul(x,y,t) = dic1)(

Lat)'

1

Figure 10 Asymmetrical plane oscillatory wake.

The boundary condition for x.0 is in this case is given as.

u1(x=0,y) = A(t)-1-211(y) - H(y+L/2) - H(y-L12)] (80) It is the same set of equations that governs the development of an asymmetrical wake as for the plane wake. The only difference is the boundary condition. Using the same argumentation

as before, we can show that the Fourier transformed velocity in x- and y-direction is given

as.

+ (1)( LI2-y )3 Laminar flow

2Vv(t-t1) (78)

L/2 -y

) + 0(

I

2VE,(t-t,)

argument in the delta function, and ti is the Turbulent flow

g(ti;x,t) = x - (g(t) - g(t)) = 0

(79)

The above solution can be extended to an asymmetrical plan oscillatory wake, shown in

Figure 10.

(41)

- cos(sL/2)

+(v +ipU = 2.,4(t)-U:(

is

The right hand side of equation (81) is the Fourier transform of the boundary condition for x=0 in y-direction. This equation is very similar to the governing equation for the plane

oscillatory wake, and the solutions are also very similar. Using same argumentation as before

we get the following solution for the Fourier transformed velocity. poeo_go)i 1 r-t 1-cos(sD2)

A(ti) 1(ti)-e it "-`')-eu'cls]dri (82)

is The last integral has to be investigated more carefully.

1(y ,v

=r-[

1

-(sLI2) -v tom

1(y,v =

-

sL/2 icos(sy) + isin(sy)]ds

tL

it is is

(y,v 1 r-e sin(sy) .ds 1

r-

cos(sL/2)-sin(sy)

n It

) 1 ro( y +La ) y -LI2 )1

1(y ,v ,t - <I?( -Y

2orT(77) 2

2v(r-r')

2\Ar (t -e)

The last part of this integral is very similar to the one we solved in the steady plane wake

case. The solution is given by a simple variable transformation. The solution of the first part of the integral is given in Gracishteyn & Ryzhik (1980) page 495.

°°'!(y ,v ,t,tr)v-vrt("/ vuY dsldt (84)

The final solution is then given by the same argumentation as before, and we get the

following expression for the development of an asymmetrical non-steady wake.

A(t)[(

Y ) 211v (t A(t I)iO( Y 211E,(t -1) 1 LI2+y

L/2-- [0(

) + cb( y )]] Laminar 2 217(7--0 2\iv (t -t1)

) -

L/2 +Y ) + 41)( Li2 -y ] Turbulent 2 2vlEt(r 21/e, (t (81) (83) (85)

N is the number of zero points in the delta function, and is the corresponding zero point. g(t = x - (g(t) - g(t ,)) = 0 (86)

A(t) can be an arbitrary function of t as long as it'samplitude is small compared to the free

flow.

The asymmetrical solution shown above can also be found directly from equation (78) by two

simple variable transformations. Dividing the asymmetrical flow into two separate plane flows, see Figure 11, we get the followinginitial condition:

= ut(x,y,t) =

=

+ A(e)-g '(h-e

-t)

(42)

y2 + 11 4)(x=0,y,t) =

- lyi

4 u2)(x=0,y,t) = - 1y21) 4

The solution is then given by changing L12 to L14, y to, yr or y, and a.U_(t) to' MOin equation (78).

Ay

X

^

Figure 11 An asymmetrical wake flow can be divided into two separate plane wake

flows.

Ay

1)

2)

=

-=

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